eye lighting + export image/video problem
A lot can be wrong with the particles to make it look bad in render. First thing I would look into to get a grip of what's going on is "if the particle system - the hair is baked". If not bake it.
As for the eyes, try to uncheck shadow in Object properties Cycles Settings of the cornea object with glass material. If that wont work, try to experiment with mixing glossy with transparent shader for cornea instead of glass. If mixed in right proportions it will let some more light into pupil object. You can also try to work something out using Light Path node to connect it's Is Shadow Ray with Mix Shader node to mix Transparent with Glass in a way that makes shadow of the object invisible. Give it a try, probably one of this ways will work for you.